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研究生：	黃御庭 Huang, Yu-Ting
論文名稱：	札可哈羅夫系統和量子札可哈羅夫系統的孤立子 The Solitary Solutions for Two Systems :The Zakharov System and The Quantum Zakharov System
指導教授：	方永富 Fang, Yung-fu
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2017
畢業學年度：	105
語文別：	英文
論文頁數：	43
中文關鍵詞：	孤立子解、札可哈羅夫系統、量子札可哈羅夫系統、F擴充法
外文關鍵詞：	soliton, Zakharov system, quantum Zakharov system, F-expansion method.
相關次數：	點閱：89 下載：8
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這篇論文中，我主要把三篇論文做詳細整理，分別是
(1) Wang Ming-Liang, Wang Yue-Ming和Zhang Jin-Liang的「兩個非線性波方程的周期解」。
(2) S.A. El-Wakil和M.A. Abdou所做的「兩個非線性物理模型的新旅行波解」。
(3) Fernando Haas, Leonardo Geissier Garcia和Joao Goedert所做的「量子札可哈羅夫系統」。
第一篇，作者們主要在找札可哈羅夫系統的孤立子解，一開始先假設我們想要的孤立子解。並且使用F-expansion方法，將孤立解改變成另外一種解的形式，並帶入方程式。根據Riccati方程調控参數來操作Maple數學軟體，來找到我們想要的孤立子解。
第二篇，在討論量子札可哈羅夫系統的孤立子解，運用類似上面的步驟，找出我們想要的孤立子解。第一二篇，除了方程差量子外，假設一開始的孤立子解方式也不同，並根據不同的Riccati方程來找想要的解。但是這篇的數據有不少打字錯誤，我把數據做更正，將結果代入方程並不滿足，因為這些解是不是正確的解，只是近似的孤立子解。
第三篇，討論量子札可哈羅夫系統在隔熱極限和半經典極限下，並運用變分法，得到許多代數方程式去求解，再估計出H ̅的範圍，來找尋我們想要的孤立子解(近似的)。

In this thesis, we mainly study the three paper as follows :
(1) 'The periodic wave solutions for two systems of nonlinear wave equations' authored by
Wang, M.L., Wang, Y.M. and Zhang, J.L. ( the F-expansion method ).
(2) 'New exact travelling wave solutions of two nonlinear physical models' authored by S.A.
El-Wakil and M.A. Abdou ( the improved F-expansion method ).
(3) 'Quantum Zakharov Equations" authored by Fernando Haas, Leonardo Geissier Garcia
and Joao Goedert ( the variational method ).
Firstly, the authors are mainly looking for solitary solutions of the Zakharov system (Z) and start presuming a specific form of solitary solutions. They use 'the F-expansion method' to transform solitary solutions into another formal solution, and put in system. They use the Maple to obtain algebraic equation that are coefficients for solitary solutions.
Secondly, they also discuss solitary solutions of the quantum Zakharov system (QZ), and use 'the improved F-expansion method'. It use a different Riccati equation to find desired solutions. And we take it to (Z) that isn't satisfied. We fill up the details which are skipped in the discussion of the paper. We also correct some typos and minor mistakes in the paper.
Thirdly, they also set out solitary solutions of (QZ) in the adiabatic limit and semiclassical limit. And they applied 'the variational method' to find the algebraic equations for solitons. In next, we need to estimate the range of H ̅ which is related to H. Respectively, they discuss small and large H ̅. Thus, we can smear out new solitary solutions of (QZ).

目錄 

摘要/ Abstract ..................................................................................................... I 

誌謝 ................................................................................................................... II 

Table of Contents ............................................................................................. III 

表目錄 .............................................................................................................. IV 

圖目錄 .............................................................................................................. IV 

Chapter 1 Introduction ...................................................................................... 1 

1.1 Background of the soliton ........................................................................... 2 

1.2 The Zakharov system ................................................................................. 3 

1.3 The quantum Zakharov system .................................................................. 3 

Chapter 2 The first method for the Zakharov system ....................................... 4 

2.1 The F-expansion method ........................................................................... 4 

2.2 The F-expansion method for (Z) ................................................................ 5 

2.3 Jacobi elliptic function and the Riccati equation ........................................ 8 

2.4 Solitary solutions ....................................................................................... 15 

Chapter 3 The second method for the quantum Zakharov system ................. 18 

3.1 The improved F-expansion method .......................................................... 18 

3.2 The improved F-expansion method for (QZ) ............................................ 19 

3.3 The Riccati equation and Cardano method .............................................. 25 

3.4 Solitary solutions ....................................................................................... 30 

Chapter 4 The third method for the quantum Zakharov system ..................... 33 

4.1 The variational method for (QZ) ............................................................... 33 

4.2 The variational method to Euler–Lagrange equation ............................... 39 

4.3 Solitary solutions ...................................................................................... 41 

Conclusion ..................................................................................................... 42 

References .................................................................................................... 43 

表目錄 
Table 1 ........................................................................................................... 12 
Table 2 ........................................................................................................... 14 
Table 3 ........................................................................................................... 14 
Table 4 ........................................................................................................... 28 
圖目錄 
Figure 1 ........................................................................................................... 2 
Figure 2 ........................................................................................................... 5 
Figure 3 ........................................................................................................... 9 
Figure 4 ......................................................................................................... 10 
Figure 5 ......................................................................................................... 15 
Figure 6 ......................................................................................................... 15 
Figure 7 ......................................................................................................... 16 
Figure 8 ......................................................................................................... 16 
Figure 9 ......................................................................................................... 17 
Figure 10 ....................................................................................................... 17 
Figure 11 ....................................................................................................... 26 
Figure 12 ....................................................................................................... 31 
Figure 13 ....................................................................................................... 31 
Figure 14 ....................................................................................................... 31 
Figure 15 ....................................................................................................... 32 
Figure 16 ....................................................................................................... 32 
Figure 17 ....................................................................................................... 32 
Figure 18 ....................................................................................................... 39 
Figure 19 ....................................................................................................... 41 
Figure 20 ....................................................................................................... 42 
Figure 21 ....................................................................................................... 42 

                                    

M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering transform.
London Mathematical Society Lecture Note Series. (1991), 149.

S.A. El-Wakil and M.A. Abdou, New exact travelling wave solutions of two nonlinear physical models. ScienceDirect. Nonlinear Analysis 68 (2008), 235-245.

Hua Gao, Symbolic computation and new exact travelling solutions for the (2+1)-dimensional zoomeron equation.
International Journal of Modern Nonlinear Theory and Application. 3 (2014), 23-28.

M.V. Goldman, Auxiliary Function Method for Nonlinear Partial Differential Equations. Rev. Mod. Phys. 56 (1984), 709.

M.V. Goldman, Strong turbulence of plasma waves. Rev. Mod. Phys. 56 (1984), 709.

F. Haas, L.G. Garcia, J. Goedert, and G. Manfredi, Quantum ion-acoustic waves. Phys. Rep. 10 (2003), 3858.

L.G. Garcia, F. Haas, L.P L. De Oliveira and J. Goedert, Modified Zakharov equations for plasmas with a quantum correction. Phys. Rep. 10 (2003), 0410257.

Fernando Haas, Leonardo Geissler Garcia and João Goedert, Quantum Zakharov Equations. Fourth International Winter Conference on Mathematical Methods in Physics. 35 (2004), 09-13.

Bor-Jian Lee, Calculus of Variation -the Brachistochrone Problem. Gamma Journals. 38 (2003), 27~34.

G. Manfredi and F. Haas, Self-consistent fluid model for a quantum electron gas. Phys. Rev. B64 (2001), 075316.

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Ming-Liang Wang, Yue-Ming Wang and Jin-Liang Zhang, The periodic wave solutions for two systems of nonlinear wave equations. Chin. Phys. Soc. and IOP Publishing Ltd. Vol 12 No 12}(2003), 1341-1348.

Fuding Xie, Zhen Wang and Min Ji1, Application of Symbolic Computation in Nonlinear Differential-Difference Equations. Discrete Dynamics in Nature and Society (2009), Article ID 158142, 8 pages.

C.T. Yan, A simple transformation for nonlinear waves.
Physics Letters. A 224 (1996), 77–84.

V.E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP 35 (1962), 908-914.

Y.B. Zhou, M.L. Wang and Y.M. Wang, Auxiliary Function Method for Nonlinear Partial Differential Equations. Phys. Lett. A. 308 (2003).

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